International Journal of Mathematics and Mathematical Sciences
Volume 2007 (2007), Article ID 65947, 14 pages
doi:10.1155/2007/65947
Research Article
Nonlinear Integrodifferential Equations of Mixed Type in Banach Spaces
1Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Poznań 61-614, Poland
2University of Marketing and Management, Ostroroga 9a, Leszno 64-100, Poland
Received 27 February 2007; Revised 28 April 2007; Accepted 19 June 2007
Academic Editor: Marco Squassina
Copyright © 2007 Aneta Sikorska-Nowak and Grzegorz Nowak. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We prove two existence theorems for the integrodifferential equation of mixed type: x'(t)=f(t,x(t),∫0tk1(t,s)g(s,x(s))ds,∫0ak2(t,s)h(s,x(s))ds), x(0)=x0, where in the first part of this paper f, g, h, x are functions with values in a Banach space E and integrals are taken in the sense of Henstock-Kurzweil (HK). In the second part f, g, h, x are weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis (HKP) integral. Additionally, the functions f, g, h, x satisfy some conditions expressed in terms of the measure of noncompactness or the measure of weak noncompactness.